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Hoernle, August Frederich 

On the Bakshali manuscript 
















Separat- A bd ruck 

Verhandluugeu des VII. Internationalen Orientalisten-Congresses. 

Arische Section, S. 127 ff. 

Druck von Adolf Holzhausen, 
k. k. Hof- und Uiiiversitiits-Btichdriicker in Wien. 

Ihe manuscript which I have the honour, this morning, of 
placing before you, was found, as you will recollect, in May 1881, 
near a village called Bakhshall, lying in the Yusufzai district of 
the Peshawer division, at the extreme Northwestern frontier of 
India. It was dug out by a peasant in a ruined enclosure, where 
it lay between stones. After the find it was at once forwarded 
to the Lieutenant Governor of the Panjab who transmitted it to 
me for examination and eventual publication. 

The manuscript is written in Sharada character of a rather 
ancient type, and on leaves of birch-bark which from age have 
become dry like tinder and extremely fragile. Unfortunately, 
probably through the careless handling of the finder, it is now 
in an excessively mutilated condition, both with regard to the 
size and the number of the leaves. Their present size, as you 
observe (see Plate I), is about 6 by 3y 2 inches; their original 
size, however, must have been about 7 by 8Y 4 inches. This 
might have been presumed from the well-known fact that the old 
birck-bark manuscripts were always written on leaves of a squarish 
size. But I was enabled to determine the point by a curious fact. 
The mutilated leaf which contains a portion of the 27 Vj sutra, 
shows at top and bottom the remainders of two large square 

4 R. Hoernle. 

figures, such as are used in writing arithmetical notations. These 
when completed prove that the leaf in its original state must 
have measured approximately 7 by 8 l / 4 inches. The number of 
the existing leaves is seventy. This can only be a small portion 
of the whole manuscript. For neither beginning nor end is pre- 
served; nor are some leaves forthcoming which are specifically 
referred to in the existing fragments. 1 ) From all appearances, 
it must have been a large work, perhaps divided in chapters 
or sections. The existing leaves include only the middle portion 
of the work or of a division of it. The earliest sutra that I have 
found is the 9 th ; the latest is the 57 th . The lateral margins 
which usually exhibit the numbering of the leaves are broken 
off. It is thus impossible even to guess what the original number 
of the leaves may have been. 

The leaves of the manuscript, when received by me, were 
found to be in great confusion. Considering that of each leaf 
the top and bottom (nearly two thirds of the whole leaf) are lost, 
thus destroying their connection with one another, it may be ima- 
gined that it was no easy task to read and arrange in order 
the fragments. After much trouble I have read and transcribed 
the whole, and have even succeeded in arranging in consecutive 
order a not inconsiderable portion of the leaves containing 
eighteen sutras. The latter portion I have also translated in 

v/ The beginning and end of the manuscript being lost, both 
the name of the work and of its author are unknown. The sub- 
ject of the work, however, is arithmetic. It contains a great 
variety of problems relating to daily life. The following are 
examples. ,In a carriage, instead of 10 horses, there are yoked 5; 
the distance traversed by the former was one hundred, how much 
will the other horses be able to accomplish ?' Te following is 
more complicated: ,A certain person travels 5 yojanas on the 

') Thus at the end of the 10 th sutra, instead of the usual explanation, 
there is the following note: evam sutram | dviflya patre mvaritasti. The leaf 
referred to is not preserved. 

On the Bakhshali Manuscript. 5 

first day, and 3 more on each succeeding day; another who 
travels 7 yojanas on each day, has a start of 5 days; in what 
time will they meet?' The following is still more complicated: 
'Of 3 merchants the first possesses 7 horses, the second 9 ponies, 
the third 10 camels; each of them gives away 3 animals to be 
equally distributed among themselves; the result is that the value 
of their respective properties becomes equal; how much was the 
value of each merchant's original property, and what was the 
value of , each animal?' The method prescribed in the rules for 
the solution of these problems is extremely mechanical, and re- 
duces the labour of thinking to a minimum. For example, the 
last mentioned problem is solved thus: 'Subtract the gift (3) se- 
verally from the original quantities (7, 9, 10). Multiply the re- 
mainders (4, 6, 7) among themselves (168, 168, 168). Divide each 
of these products by the corresponding remainder (^ ^, ^-p-). 
The results (42, 28, 24) are the values of the 3 classes of animals. 
Being multiplied with the numbers of the animals originally pos- 
sessed by the merchants (42 . 7, 28 . 9, 24 . 10), we obtain the 
values of their original properties (294, 252, 240). The value of 
the properts of each merchant after the gift is equal (262, 262, 
262).' The rules are expressed in very concise language, but are 
fully explained by means of examples. Generally there are two 
examples to each rule (or sutra), but sometimes there are many; 
the 25 th sutra has no less than 15 examples. The rules and 
examples are written in verse; the explanations, solutions and 
all the rest are in prose. The metre used is the shloka. 

The subject-matter is divided in sutras. In each sutra the 
matter is arranged as follows. First comes the rule, and then 
the example, introduced by the word tada. Next, the example 
is repeated in the form of a notation in figures, which is called 
sthapana. This is followed by the solution which is called karana. 
Finally comes the proof, called pratyaya. This arrangement and 
terminology differ somewhat from those used in the arithmetic 
of Brahmagupta and Bhaskara. Instead of simply sutra, the latter 
use the term karana-sutra. The example they call uddeshaka or 
udaharana. For sthapana they say nyasa. As a rule they give 

6 R. Hoernle. 

no full solution or proof, but the mere answer to the problem. 
Occasionally a solution is given, but it is not called karana. 

The system of notation used in the Bakhshali arithmetic 
is much the same as that employed in the arithmetical works of 
Brahmagupta and Bhaskara. There is, however, a very impor- 
tant exception. The sign for the negative quantity is a cross (+). 
It looks exactly like our modern sign for the positive quantity, 
but is placed after the number which it qualifies. Thus V l + 
means 12 7 (i. e. 5). This is a sign which I have not met with 
in any other Indian arithmetic; nor so far as I have been able 
to ascertain, is it known in India at all. The sign now used 
is a dot placed over the number to which it refers. Here, there- 
fore, there appears to be a mark of great antiquity. As to its 
origin I am unable to suggest any satisfactory explanation. I 
have been informed by Dr. Thibaut of Benares, that Diophantus 
in his Greek arithmetic uses the letter i| (short for Xetyi?) reversed 
(thus fj>), to indicate the negative quantity. There is undoubtedly 
a slight resemblance between the two signs; but considering 
that the Hindus did not get their elements of the arithmetical 
science from the Greeks, a native origin of the negative sign 
seems more probable. It is not uncommon in Indian arithmetic 
to indicate a particular factum by the initial syllable of a word 
of that import subjoined to the terms which compose it. Thus 
addition may be indicated by yu (short for yuta\ e. q. f \ w 
means 5 + 7 (c. e. 12). In the case of substraction or the ne- 
gative quantity rina would be the indicatory word and ri the 
indicatory syllable. The difficulty is to explain the connection 
between the letter ri (^?) and the symbol +. The latter very 
closely resembles the letter k (^j) in its ancient shape (+) as 
used in the Ashoka alphabet. The word kana or kaniyas which 
had once occurred to me, is hardly satisfactory. 

A whole number, when it occurs in an arithmetical opera- 
tion, as may be seen from the above given examples, is in- 
dicated by placing the number 1 under it. This, however, is a 
practice which is still occasionally observed in India. It may 
be worth noting that the number one is always designated by 

On the Bakhslmll Manuscript. 7 

the word rupa- 1 ) thus sarupa or rupadhika 'adding one', rupona 
'deducting one'. The only other instance of the use of a sym- 
bolic numeral word is the word rasa for six which occurs once 
in an example in sutra 53. 

The following statement, from the first example of the 
25 th Sutra, affords a good example of the system of notation 
employed in the Bakhshali arithmetic: 

. J | } bha 32 

1 3+ 3+ 3+ 

phalam 108 

Here the initial dot is used very much in the same way as we 
use the letter x to denote the unknown quantity the value of 
which is sought. The number 1 under the dot is the sign of the 
whole (in this case, unknown) number. A fraction is denoted 
by placing one number under the other without any line of 
separation; thus J is f, i. e. one-third. A mixed number is shown 
by placing the three numbers under one another; thus 1 is 1 -f- f 
or If, i. e. one and one-third. Hence i + means 1 f (i. e. f). 
Multiplication is usually indicated by placing the numbers side 
by side; thus I g V | phalam 20 means f X 32 = 20. Similarly 
| + j + ] + means f XIX! or (f) 3 , c. e. . Bha is an abbre- 
viation of bhaga 'part' and means that the number preceding 
it is to be divided. Hence ] + i + i + bha means ". The whole 
statement, therefore, 

1 3+ 3+ 3+ bha 32 I phalam 108 

means ~ X 32 = 108, and may be thus explained: 'a certain 
number is found by dividing with -j^ and multiplying with 32; 
that number is 108'. 

The dot is also used for another purpose, namely as one 
of the ten fundamental figures of the decimal system of notation 

*) This word was at first read by me upa. The reading rupa was sug- 
gested to me by Professor A. Weber, and though not so well agreeing with 
the manuscript characters, is probably the correct one. 


8 R. Hoernle. 

or the zero (0123456789). It is still so used in India for 
both purposes, to indicate the unknown quantity as well as the 
naught. With us the dot, or rather its substitute the circle (), 
has only retained the latter of its two intents, being simply the 
zero figure, or the 'mark of position' in the decimal system. 
The Indian usage, however, seems to show, how the zero arose 
and that it arose in India. The Indian dot, unlike our modern 
zero, is not properly a numerical figure at all. It is simply a 
sign to indicate an empty place or a hiatus. This is clearly 
shown by its name sliunya 'empty". The empty place in an 
arithmetical statement might or might not be capable of being 
filled up, according to circumstances. Occurring in a row of 
figures arranged decimally or according to the Value of position", 
the empty place could not be filled up, and the dot therefore 
signified 'naught', or stood in the place of the zero. Thus the 
two figures 3 and 7, placed in juxtaposition (37) mean 'thirty 
seven', but with an 'empty space' interposed between them (3 7), 
they mean 'three hundred and seven'. To prevent misunder- 
standing the presence of the 'empty space' was indicated by a 
dot (3*7), or by what in now the zero (307). On the other hand, 
occurring in the statement of a problem, the 'empty place' could 
be filled up, and here the dot which marked its presence, signi- 
fied a 'something" which was to be discovered and to be put in 
the empty place. In the course of time, and out of India, the 
latter signification of the dot was discarded; and the dot thus 
became simply the sign for 'naught' or the zero, and assumed 
the value of a proper figure of the decimal system of notation, 
being the 'mark of position'. In its double signification which 
still survives in India, we can still discern an indication of that 
country as its birth place. 

Regarding the age of the manuscript am unable to offer 
a very definite opinion. The composition of a Hindu work on 
arithmetic, such as that contained in the Bakhshall MS. seems 
necessarily to presuppose a country and a period in which Hindu 
civilisation and Brahmanical learning flourished. Now the country 
in which Bakhshall lies and which formed part of the Hindu 

kingdom of Kabul, was early lost to Hindu civilisation through 
the conquests of the Muhammedan rulers of Ghazni ; and espe- 
cially through the celebrated expeditions of Mahmiid, towards 
the end of the 10 th and the beginning of the 11 th centuries A. D. 
In those troublous times it was a common practice for the learned 
Hindus to bury their manuscript treasures. Possibly the Bakhshall 
MS. may be one of these. In any case it cannot well be placed 
much later than the 10 th century A. D. It is quite possible that 
it may be somewhat older. The Sharada characters used in it, 
exhibit in several respects a rather archaic type, and afford 
some ground for thinking that the manuscript may perhaps go 
back to the 8 th or 9 th century. But in the present state of our 
epigraphical knowledge, arguments of this kind are always some- 
what hazardous. The usual form, in which the numeral figures 
occur in the manuscript are the following: 

1 2 3 4567890 

Quite distinct from the question of the age of the manu- 
script is that of the age of the work contained in it. There is 
every reason to believe that the Bakhshall arithmetic is of a 
very considerably earlier date than the manuscript in which it 
has come down to us. I am disposed to believe that the com- 
position of the former must be referred to the earliest centuries 
of our era, and that it may date from the 3 d or 4 th century 
A. D. The arguments making for this conclusion are briefly the 

In the first place, it appears that the earliest mathematical 
works of the Hindus were written in the Shloka measure; *) 
but from about the end of the 5 th century A. D. it became the 
fashion to use the Arya measure. Aryabhatta c. 500 A. D., Va- 
raha Mihira c. 550, Brahmagupta c. 630, all wrote in the latter 
measure. Not only were new works written in it, but also Shloka 
works were revised and recast in it. Now the Bakhshall arith- 

See Professor Kern's Introduction to Varaha Mihira. 

10 R. Hoernle. 

metic is written in the Shloka measure; and this circumstance 
carries its composition back to a time anterior to that change 
of literary fashion in the 5 th century A. D. 

In the second place, the Bakhshall arithmetic is written 
in that peculiar language which used to be called the 'Gatha 
dialect', but which is rather the literary form of the ancient 
Northwestern Prakrit (or Pali). It exhibits a strange mixture of 
what we should now call Sanskrit and Prakrit forms. As shown 
by the inscription (e. g., of the Indoscythian kings in Mathura) 
of that period, it appears to have been in general use, in North- 
western India, for literary purposes till about the end of the 
3 d century A. D., when the proper Sanskrit, hitherto the language 
of the Brahmanic schools, gradually came into general use also 
for secular compositions. The older literary language may have 
lingered on some time longer among the Buddhists and Jains, 
but this would only have been so in the case of religious, not 
of secular compositions. Its use, therefore, in the Bakhshall arith- 
metic points to a date not later than the 3 d or 4 th century A. D. 
for the composition of that work. 

In the third place, in several examples, the two words 
dlndra and dramma occur as denominations of money. These 
words are the Indian forms of the latin denarius and the greek 
drachme. The former, as current in India, was a gold coin, the 
latter a silver coin. Golden denarii were first coined at Rome 
in 207 B. C. The Indian gold pieces, corresponding in weight 
to the Roman gold denarius, were those coined by the Indoscy- 
thian kings, whose line beginning with Kadphises, about the 
middle of the 1 st century B. C., probably extended to about the 
end of the 3 d century A. D. Roman gold denarii themselves, as 
shown by the numerous finds, were by no means uncommon in 
India, in the earliest centuries of our era. The gold dinars most 
numerously found are those of the Indoscythian kings Kanishka 
and Huvishka, and of the Roman emperors Trajan, Hadrian 
and Antonius Pius, all of whom reigned in the 2 nd century A. D. 
The way in which the two terms are used in the Bakhshall 
arithmetic seems to indicate that the gold dinara and the silver 

dramma formed the ordinary currency of the day. This circum- 
stance again points to some time within the three first . centuries 
of the Christian era as the date of its composition. 

A fourth point, also indicative of antiquity which I have 
already adverted to, is the peculiar use of the cross (-J-) as the 
sign of the negative quantity. 

There is another point which may be worth mentioning 
though I do not know whether it may help in determining the 
probable date of the work. The year is reckoned in the Bakhshali 
arithmetic as consisting of 360 days. Thus in one place the fol- 
lowing calculation is given: Tf in fff of a year 2982 -J-J-J- is 
spent, how much is spent in one day?' Here it is explained 
that the lower denomination (adha-ch-chheda) is 360 days, and 
the result (phala) is given as ^ (i. e. nfft^iii__). 

In connection with this question of the age of the Bakhshali 
work, I may note a circumstance which appears to point to a 
peculiar connection of it with the Brahmasiddhanta of Brahma- 
gupta. There is a curious resemblance between the 50 th sutra of 
the Bakhshali arithmetic, or rather with the algebraical example 
occurring in that sutra, and the 49 th sutra of the chapter on 
algebra in the Brahmasiddhanta. In that sutra, Brahmagupta 
first quotes a rule in prose, and then adds another version of 
it in the Arya measure. Unfortunately the rule is not preserved 
in the Bakhshali MS., but as in the case of all other rules, it 
would have been in the form of a shloka and in the North- 
western Prakrit (or 'Gatha dialect'). Brahmagupta in quoting it, 
would naturally put it in what he considered correct Sanskrit 
prose, and would then give his own version of it in his favourite 
Arya measure. I believe it is generally admitted that Indian 
arithmetic and algebra, at least, is of entirely native origin. While 
siddhanta writers, like Brahmagupta and his predecessor Arya- 
bhatta, might have borrowed their astronomical elements from 
the Greeks or from books founded themselves on Greek science, 
they took their arithmetic from native Indian sources. Of the 
Jains it is well known that they possess astronomical books of 
a very ancient type, showing no traces of western or Greek 

12 R. Hoernle. 

influence. In India arithmetic and algebra are usually treated 
as portions of works on astronomy. In any case it is impossible 
that the Jains should not have possessed their own treatises on 
arithmetic when they possessed such on astronomy. The early 
Buddhists, too, are known to have been proficients in mathe- 
matics. The prevalence of Buddhism in Northwestern India, in 
the early centuries of our era, is a well known fact. That in 
those early times there were also large Jain communities in those 
regions is testified by the remnants of Jain sculpture found near 
Mathura and elsewhere. From the fact of the general use of the 
Northwestern Prakrit (or the 'Gatha dialect') for literary purposes 
among the early Buddhists it may reasonably be concluded that 
its use prevailed also among the Jains between whom and the 
Buddhists there was so much similarity of manners and customs. 
There is also a diffusedness in the mode of composition of the 
Bakhshall work which reminds one of the similar characteristic 
observed in Buddhist and Jain literature. All these circumstances 
put together -seem to render it probable that in the Bakhshall MS. 
we have preserved to us a fragment of an early Buddhist or 
Jain work on arithmetic (perhaps a portion of a larger work on 
astronomy) which may have been one of the sources from which 
the later Indian astronomers took their arithmetical information. 
These earlier sources, as we know, were written in the shloka 
measure, and when they belonged to the Buddhist or Jain lite- 
rature, must have been composed in the ancient Northwestern 
Prakrit. Both these points are characteristics of the Bakhshall 
work. I may add that one of the reasons why the earlier works 
were, as we are told by tradition, revised and rewritten in the 
Arya measure by later writers such as Brahmagupta, may have 
been that in their time the literary form (Gatha dialect) of the 
Northwestern Prakrit had come to be looked upon as a barba- 
rous and ungrammatical jargon as compared with their own classi- 
cal Sanskrit. In any case the Buddhist or Jain character of the 
Bakhshall arithmetic would be a further mark of its high antiquity. 
Throughout the Bakhshall arithmetic the decimal system 
of notation is employed. This system rests on the principle of 

On the Bakhshall Manuscript. 13 

the 'value of position' of the numbers. It is certain that this prin- 
ciple was known in India as early as 500 A. D. There is no 
good reason why it should not have been discovered there con- 
siderably earlier. In fact, if the antiquity of the Bakhshall arith- 
metic be admitted on other grounds, it affords evidence of an 
earlier date of the discovery of that principle. As regards the 
zero, in its modern sense of a 'mark of position' and one of the 
ten fundamental figures of the decimal system (0123456789), 
its discovery is undoubtedly much later than the discovery of the 
Value of position 1 . It is quite certain, however, that the appli- 
cation of the latter principle to numbers in ordinary writing 
would have been nearly impossible without the employment of 
some kind of 'mark of position', or some mark to indicate the 
'empty place 1 (shunya). Thus the figure 7 may mean either 'seven' 
or 'seventy' or 'seven hundred' according as it be or be not 
supposed to be preceded by one (7 or 70) or two (7 or 700) 
'empty places'. Unless the presence of these 'empty places' or 
the 'position' of the figure 7 be indicated, it would be impossible 
to read its 'value' correctly. Now what the Indians did, and in- 
deed still do, was simply to use for this purpose the sign which 
they were in the habit of using for the purpose of indicating 
any empty place or omission whatsoever in a written composi- 
tion; that is the dot. It seems obvious from the exigencies of 
writing that the use of the well known dot as the mark of an 
empty place must have suggested itself to the Indians as soon 
as they began to employ their discovery of the principle of 
'value position' in ordinary writing. In India the use of the dot 
as a substitute of the zero must have long preceded the disco- 
very of the proper zero, and must have been contemporaneous 
with the discovery of that principle. There is nothing in the 
Bakhshall arithmetic to show that the dot is used as a proper 
zero, and that it is any thing more than the ordinary 'mark of 
an empty place'. The employment, therefore, of the decimal 
system of notation, such as it is, in the Bakhshall arithmetic is 
quite consistent with the suggested antiquity of it. 

14 R- Hoernle. 

I have already stated that the Bakhshall arithmetic is written 
in tho so-called 'Gatha dialect 1 , or in that literary form of the 
Northwestern Prakrit, which preceded the employment, in secu- 
lar composition, of the classical Sanskrit. Its literary form con- 
sisted in what may be called (from the Sanskrit point of view) 
an imperfect sanskritisation of the vernacular Prakrit. Hence it 
exhibits at every turn the peculiar characteristics of the under- 
lying vernacular. The following are some specimens of ortho- 
graphical peculiarities. 

Insertion of euphonic consonants: of m, in eka-m-ekatvam, 
bhritako - m - ekapanditah of r, in tri - r - dshlti, labhate-r- 

Insertion of s: vibhdktam-s-uttare, Ksiyate-s-traya. This is 
a peculiarity not elsewhere known to me, either in Pra- 
krit or in Pali. 

Doubling of consonants: in compounds, prathama-d-dhdnttt, 
eka-s-samkhyd; in sentences, yadi-s-sadbhi 7 ete-s-sama- 

Peculiar spellings: trinslia or trinsha for trimshat. The 
spelling with the guttural nasal before sh occurs only 
in this word; e. q., chatvaliihsha 40. Again ri for ri in 
tridine, kriyate, vimishritam, krindti; and ri for ri in 
rinam, dristah. Again katthyatam for kathyatdm. Again 
the jihvdmullyo and the upadhmdniya are always used 
before gutturals and palatals respectively. 

Irregular sandhi: ko so ra for kah sa ra, dvayo kechi 
for dvayah k, dvayo cha for dvayash cha, dvibhi kri 
for dvibhih kri, adyo vi* for ddyor vi, vivaritdsti for 
vivaritam asti. 

Confusion of the sibilants: sh for s, in shasti 60, mdshako; 
s for sh, in dashdmsha, visodliayet, sesam; sh for s, in sd- 
shyam, sdsyatdm; s for sh, in esa ,this^. 

Confusion of n and n: utpanna. 

Elision of a final consonant: bhdjaye, kechi for bhdjayet, 

On the Bakhshali Manuscript. 15 

Interpolation of r: hrlnam for hlnam. 

The following are specimens of etymological and syntactical 

Absence of inflection: nom. sing, masc., esha sd rdshi for ra- 
shih (s. 50), gavdm vishesa kartavyam for vishesah (s. 51). 
Nom. plur., sevya santi for sevydh (s. 53). Ace. plur., 
dlnara dattavdn for dmdrdn (s. 53). 

Peculiar inflection: gen. sing., gatisya for gateh (s. 15); 
atm. for parasm., dr jay ate for arjayati r he earns 1 (s. 53); 
parasm. for atm., vikrindti for vikrmlte 'he sells 1 (s. 54). 

Change of gender: masc. for neut., mida for mulani (s. 55); 
neut. for masc., vargam for vargah (s. 50); neut. for fern., 
yutim cha kartavyd for yutish (s. 50). 

Exchange of numbers: plur. for sing., (bhavet) Idbhdh for 
Idbhah (s. 54). 

Exchange of cases: ace. for nom., dvitlyam pamchadivase 
rasam ar jay ate for dmtiyah (s. 53); ace. for instr., ksayam 
samgunya for ksayena (s. 27); ace. for loc., kim kdlam 
for kasmin kale (s. 52); instr. for loc., anena kdlena for 
asmin kale (s. 53); instr. for nom., prathamena dattavdn 
for prathamo (s. 53), or ekena ydti for eko (s. 15); loc. 
for instr., prathame dattd for prathamena (s. 53), or ma- 
nave grihltam for mdnavena (s. 57); gen. for dat., dvi- 
tlyasya dattd for dvitiydya (s. 53). 

Abnormal concord: incongruent cases, ay am praste for as- 
min (s. 52); incongruent numbers, esha Idbhdh for Id- 
bhah (s. 54) rdjaputro kechi for rdjaputrah (s. 53); incon- 
gruent genders, sd kdlam for tat kdlam (s. 52), vishesa 
kartavyam for kartavyah (s. 51), sd rdshih for sa (s. 50), 
kdryam sthitah for sthitam (s. 14). 

Peculiar forms: nivarita for nivrita, drja for drjana, divad- 
dha 'one and one-half, chatvdlimsha 40, pamchdshama 
50 th , chaupamchdshama 54 th , chaturdshUi 84, tri-r-dshlti 
83, etc. 

The following extracts may serve as specimens of the text. 

16 R. Hoernle. 

Sutram | 

Adyor vishesadvigunam chayashuddhi vibhajitam 
Hup&dhikam tathd &alam gatisasyam tada bhavet || 
tada I 

Dvayaditrichayash chaiva dvic/zo/ar?/adikottarah | 
Dvayo cha bhavate pamtha kena kalena sasyatam || 
sthapanam kriyate | esdm || a j || u I \\ pa \ \\ dvi | a I || u I \\ pa 

karanam | adyor vishesa 

. ta dvi 2 . 


a } 

a \ 

u J 

pa ; 
pa { 

dha ; 
1 dha 1 

karanam | 
adi d \ 10 

adyor vishesam 
vishesa 5 cha- 

yashuddhi chayam 6 3 shuddhi 3 adishesa 5 dvigunam 10 utta- 
ravishesa 3 vibhaktam " sarupam esa padaih anena ka^ena sa- 
madhana bhavanti || pratyayam || ruponakaranena phalam ^ v { 65 || 
Asthadasashamasutram 18 || -^ I 

IdanTm suvarnaksayam vaksyami yasyedam sutram | 

Siitram | 

Ksayam samgunya kanakas tadyuti-b-bhajayf tatah ] 
Samyutair eva kanakair ekaikasya ksayo hi sah | 
tada || 

Ekadvitrichatussamkhya, suvarna masakai rinai | 
Ekadvitrichatussamkhi/a/ rahita samabhagatam || 
sthapanam kriyate | esam J \ + \\ l + \\ l + \\ t + [ karanam | ksayam 
samgunya kanakadibhi ksayena samgunya jatam 1 | 4 9 16 | 
tadyuti esa yati 30 kanaka yuti 10 anena bhaktva labdham 

On the Bakh shall Manuscript. 


10 ; so ; i 
i ; i j i 


mase 3 

: 10 ; so : 2 
; i ; i I i 

\ pha 

mdse i 


30 1 3 

| pha 



30 I 3 
1 1 1 

| pha 

mase \ 2 

tada || 

Ek&d.viti'ichatussamkhyd suvarna projjhita line | 

Masaka dvitritarii chaiva chatuhpariichakarariishakaih *) kim ksa- 
yarii || 

karanarii | ksayarii saihgunya kanaka esha stha- 
pyate | | \ \ j \ I \ { | -s-tadyuti-b-bhajayeta 2 ) ta- 

tah harasasye krite yutaiii | ^ | samyutaih kanakair bhaktva tada 

kanaka 10 anena bhaktaih jatarii | 6 6 3 | esha ekaikasuvarnasya 

ksayam || pY&tyayam ^-airashikena kartavya ||' 

10 163 ! 1 
1 ; 60 : 1 

'io"":""i63'"; 2 

i ; eo i i 

10 I 163 j 3 

1 I 60 I 1 



' 1 

163 j 

60 j 

4 1 -nV.0 103 

i | pna eoo 


shrunushva me ] 

Kramena dvaya masadi uttare ekahmatam | 
Suvarnam me tu sammishrya katthyatam ganakottama || 
sthapanaih || 4 5 f | 5 6 f || ? + || l + \\ l + \\ i 9 + | 2 + || ! + II l + II ksa- 

yam saiiigunya jatarii 20 30 | 42 | 56 | 72 90 | 2 | 6 1 3) esaiii 
yuti 330 kana&anaiii yuti 45 I anena bhaktva labdham | 3 4 30 | parii- 
chadash abhage-sh-chheda kriyate ! phalarii | 7 she 3 | esha ekaika- 
mashakaksayaih | pratyaya trairashikena \ 4 , 5 | 330 1 \ \ phalarii 2 3 2 | 
evarii sarvesam pratyaya kartavya \\ 

Saptavimshatimasufram 27 | ^ || 

1 ) Read chatuhpamchamsham kim ksayam, metri causa. 

2 ) liead bhajayet. 

3 ) Here | 12 | is omitted in the text, by mistake. 


K. Hoornle. 

Sutram | 

Ahadravyaharashauta *) tadvishesaih vibhajayet \ 
Yallabdharii dvigunarii kalarii datta samadhana prati || 

tada || 

Tridine arjaye pamcha bhritako-m-ekapanditah | 
Dvitlyam pamchacfo'vase rasam arjayate budhah || 
Prathamena dvitiyasya sapta dattani . . tah | 
Datva samadhana jata kena kalena katthyatam I 

I! 3 ru II 5 ru II . . . m ....... m ha?'mshauta tadvishesam 

anena kalena samadhana bhavanti || pratyaya trairashike kriyate 

pha 50 

P ha 

prathame dvitlyasya-s-sapta datta | 7 she- 
sam 43 || 43 | 43 ete samadhana jata || 


-Kojaputro dvayo kechi nripati-s-sevya santi vaih | 
M-ekasyahne dvaya-s-sadbhaga 2 ) dvitiyasya divarddhakaih || 
Prathamena dvitiyasya dasha dinara dattavan | 
Kena kalena samatarii ganayitva vadashu me || 

karanam | aha dravyavishesam cha tatra 

2 I dattam 

pratyayam trairashik 
tiyasya 10 datta jata 
jata 1 Sutram tripam 









prathamena dvi- 
55 1 samadhana 

hamah sutram 53 

I 3 

The 18 th Sutra. 

Let twice the difference of the two initial terms be divided 
by the difference of the (two) increments. The result augmented 
by one shall be the time that determines the progression. 

') Read haramshauta. 

2 ) Read ekasyalme dvi^adbhaga. The error appears to have been no- 

ticed by the scribe of the manuscript. 

On the Bakhshali Manuscript. 19 

First Example. 

A person has an initial (speed) of two and an increment 
of three, another has an increment of two and an initial (speed) 
of three. Let it now be determined in what time the two persons 
will meet in their journey. 

The statement is as follows: 

N I, init. term 2, increment 3, period x 
NII, 3, 2, x. 

Solution: the difference of the two initial terms (2 and 3 is 1 ; 
the difference of the two increments 3 and 2 is 1 ; twice the diffe- 
rence of the initial terms 1 is 2, and this, divided by the diffe- 
rence of the increments 1 is 2 /j , and augmented by 1 is Vj ; this is 
the period. In this time [3] they meet in their journey which is 15). 

Second Example. 

(The problem in words is wanting; it would be something 
to this effect: A earns 5 on the first and 6 more on every fol- 
lowing day; B earns 10 on the first and 3 more on every fol- 
lowing day; when will both have earned an equal amount?) 


N 1, init. term 5, increment 6, period x, possession x 
N 2, 10, 3, x, x. 

Solution: 'Let twice the difference of the two initial terms', 
etc. ; the initial terms are 5 and 10, their difference is 5. 'By the 
difference of the (two) increments'; the increments are 6 and 3; 
their difference is 3. The difference of the initial terms 5, being 
doubled, is 10, and divided by the difference of the increments 3, 
is -^, and augmented by one is . This (i. e. -^ or 4J-) is the 
period; in that time the two persons become possessed of the 
some amount of wealth. 

Proof: by the 'ruponcC method the sum of either progres- 
sion is found to be 65 (i. e., each of the two persons earns 65 
in 4y 3 days). 

20 B. Hocrnlc. 

The 27 th Sutra. 

Now I shall discuss the wastage (in the working) of gold, 
the rule about which is the following. 


Multiplying severally the parts of gold with the wastage, 
let the total wastage be divided by the sum of the parts of gold. 
The result is the wastage of each part (of the whole mass) of gold. 

First Example. 

Suvarnas numbering respectively one, two, three, four are 
subject to a wastage of masakas numbering respectively one, 
two, three, four. Irrespective of such wastage they suffer an 
equal distribution of wastage. (What is the latter?) 

The statement is as follows: 

Wastage -- 1, 2, 3, 4 masaka 
Gold 1, 2, 3, 4 suvarna. 

Solution: 'Multiplying severally the parts of gold with the 
wastage 1 , etc. 5 by multiplying with the wastage, the product 1, 
4, 9, 16 is obtained; 'let the total wastage', its sum is 30; the 
sum of the parts of gold is 10; dividing with it, we obtain 3. 
(This is the wastage of each part, or the average wastage, of 
the whole mass of gold.) 

(Proof by the rule of three is the following :) as the sum 
of gold 10 is to the total wastage of 30 masakas, so the sum 
of gold 4 is to the wastage of 12 masakas , etc. 

Second Example. 

There are suvarnas numbering one, two, three, four. There 
are thrown out the following masakas: one-half, one-third, one- 
fourth, one-fifth. What is the (average) wastage (in the whole 
mass of gold)? 
Statement : 

quantities of gold, 1, 2, 3, 4 suvarnas 
wastage y. 2 , Vs> V*? Vs masakas. 

Solution : ' Multiplying severally the parts of gold with the 
wastage', the products may thus be stated: \i^ 2 /3? 3 /j? 4 /&- 'Let 

On the Bakhshali Manuscript. 21 

the total wastage be divided' ; the division being directed to be 
made, the total wastage is -~ ; dividing 'by the sum of the parts 
of gold'; here the sum of the parts of gold is 10; being divided 
by this, the result is ||f. This is the wastage of each part of 
the whole mass of gold. 

Proof: may be made by the rule of three: as the sum of 
the parts of gold 10 is to the total wastage of ^ masaka, so 
the sum of gold 4 is to the wastage of f| masaka, etc. 

Third Example. 

(The problem in words is only partially preserved, but 
from its statement in figures and the subsequent explanation, its 
purport may be thus restored.) 

Of gold masakas numbering respectively five, six, seven, 
eight, nine, ten, quantities numbering respectively four, five, six, 
seven, eight, nine, are wasted. Of another metal numbering in 
order two masaka, etc. (i. e., two, three, four) also quantities 
numbering in order one etc. (i. e. one, two, three) are wasted. 
Mixing the gold with the alloy, best of arithmeticians, tell 
me (what is the average wastage of the whole mass of gold)? 

Statement : 

wastage: -- 4, - 5, - 6, - 7, - 8, - - 9; - 1, -2, -3, 
gold: 5, 6, 7, 8, 9, 10; 2, 3, 4. 

(Solution:) 'Multiplying severally the parts of gold with 
the wastage', the product is 20, 30, 42, 56, 72, 90, 2, 6, 12; 
their sum is 330; the sum of the parts of gold is 45; dividing 
by this we obtain ~^; this is reduced by 15 (i. e. -~); the result 
is 7 leaving y 3 (i. e. 7 l / 3 ); that is the wastage of each masaka 
(of mixed gold). 

Proof: by the rule of three : as the total gold 45 is to the 
total wastage 330, so 1 masaka of gold is to parts of wastage. 
In the same way the proof of all (the other) items is to be 
made (i. e., 45 : 330 = 5 : ^; 45 : 330 = 6 : 44; 45 : 330 ' = 
7 : ifi; 45 : 330 = 8 : ip ; 45 : 330 = 9 : 66; 45 : 330 = 10 : ^f). 

22 ft. Hoernle. 

The 53 d sutra. 

Let the portion given from the daily earnings be divided 
by the difference of the latter. The quotient, being doubled, is the 
time (in which), through the gift, their possessions become equal. 

First Example. 

Let one serving pandit earn five in three days; another 
learned man earns six in five days. The first gives seven to 
the second from his earnings; having given it, their possessions 
become equal; say, in what time (this takes place)? 

Statement N 1, | earnings of 1 day, N 2, f earnings of 
1 day; gift 7. 

Solution: 'Let the portion of the daily earnings be divided 
by the difference of the latter'; (here the daily earnings are f 
and f; their difference is VIM tne gift i g ?5 divided by the 
difference of the daily earnings 7 / 15 , the result is 15; being 
doubled, it is 30; this is the time), in which their possessions 
become equal. 

Proof: may be made by the rule of three : 3 : 5 = 30 : 50 
and 5 : 6 = 30 : 36 ; 'the first gives seven to the second' 7, 
remainder 43; hence 43 and 43 are their equal possessions. 

Second Example. 

Two Rajputs are the servants of a king. The wages of one 
per day are two and one-sixth, of the other one and one-half. The 
first gives to the second ten dinars. Calculate and tell me quickly, 
in what time there will be equality (in their possessions)? 

Statement: daily wages -^ and f; gift 10. 

Solution: 'and the daily earnings'; here (the daily earnings 
are ^ and f; their difference is f; the gift is 10; divided by 
the difference of the daily earnings |, the result is 15; being 
doubled, it is 30. This is the time, in which their possessions 
become equal). 

Proof by the rule of three: 1 : - 1 / = 30 : 65; and 1 : f - 
30:55. The first gives 10 to the second; hence 55 and 55 are 
their equal possessions. 

tin the BakhshSlI Manuscript. Jo 


1. In the text, the italicised words are conjecturally re- 
stored portions. The dots signify the number of syllables (aksara) 
which are wanting in the manuscript. The serpentine lines in- 
dicate the lines lost at the top and bottom of the leaves of the 
manuscript. In the translation the bracketed portions supply lost 
portions of the manuscript. The latter can, to a great extent, be 
restored by a comparison of the several examples. Occasionally 
words are added in brackets to facilitate the understanding of 
the passage. 

2. Sutra 18. Problems on progression. Two persons advance 
from the same point. At starting B has the advantage over A] 
but afterwards A advances at a quicker rate than B. Question : 
when will they have made an equal distance? In other words, 
that period of the two progressions is to be found, where their 
sums coincide. The first example is taken from the case of two 
persons travelling. B makes 3 miles on the first day against 
2 miles of A] but A makes 3 miles more on each succeding 
day against 5's 2 miles. The result is that at the end of the 
3 d day they meet, after each has travelled 15 miles. For A tra- 
vels 2 -f- (2 -4- 3) + (2 -f 3 + 3) = 15 miles, and B 3 -f- 
(3 + 2) -f- (3 + 2 -f- 2) = 15 miles. The second example is 
taken from the case of two traders. At starting B has the ad- 
vantage of possessing 10 dinars against the 5 of A ; but in the 
sequel A gains 6 dinars more on each day against the 3 of B. 
The result is that after 4 1 / 3 days, they possess an equal amount 
of dinars f viz. 65. 

3. Sutra 27. Problems on averages (samabhagata) . Certain 
quantities of gold suffer loss at different rates. Question: what 
is the average loss of the whole? The first problem is very 
concisely expressed; the question is understood; some words, 
like kutogatoij must be supplied to samabhagatam. 




Hoernle, August Frederich 

On the Bak shall manuscript